DualitiesHEPTH Report Summary

The project was divided in two strands. Strand 1, dealing with the AdS/CFT duality, scattering amplitudes and correlation functions. Strand 2, dealing with correspondences between four dimensional super-symmetric gauge theories and two dimensional conformal field theories. We have made substantial progress in most of our initial objectives, although some of them were adjusted in the course of the project, some proved to be too ambitious and in some we have made unexpected progress. Regarding strand 1, one of the objectives was to compute scattering amplitudes of planar MSYM to all values of the coupling, and to extend this computation to the non-planar case. This proved to be too ambitious. It is still the subject of active investigations and our group has made progress in understanding a geometrical approach. This is a perturbative approach, but its extension to non-planar amplitudes is rather natural. It is fair to say that at the moment no fully satisfactory approach exists to study amplitudes non-perturbatively.

Another objective of strand 1 was the computation of correlation functions of planar MSYM. Here we have achieved our objectives and our findings actually surpassed our expectations. We have developed a method to analytically study correlators of MSYM in the spirit of the conformal bootstrap. Namely, an approach based on symmetries and consistency conditions. This approach together with insights from Feynman computations and integrability, lead us to a wealth of new results and techniques for MSYM:

.- Techniques to study structure constants for large values of the spin..- A proof of the reciprocity principle. .- Non-perturbative bounds for anomalous dimensions and structure constants of various operators. .- Results for the anomalous dimensions of operators in a large N expansion, to order 1/N^4.

The first two are valid to all loops in perturbation theory and do not require a planar limit. The third type of result also does not require planarity, and is actually valid non-perturbatively, although its only numerical. Finally, the fourth result is the first one ever of its kind, and allows to learn about loop gravity on AdS spaces, through the AdS/CFT duality. Remarkably, much of these techniques turned out to have a much wider range of applicability and this opened the path to what is now a quite active field of research.

Regarding strand 2 we have also achieved many of our objectives. In particular:

.- We have found more examples of 2d/4d correspondences and refined existing ones, e.g. by adding defects. .- We developed machineries to exactly compute super-symmetric observables in many holographic quantum field theories, in three and five dimensions; and their geometric duals. .- We presented a powerful refined version of a related 3d/3d correspondence.

On the other hand, some of the objectives, such as understanding the elusive (2,0) 6d theory, proved to be very ambitious, and we have only made indirect progress (for instance, one of the techniques developed in strand 1 may be the best tool to study 1/N corrections to this theory).

Finally, one of the objectives of this proposal was to find common mathematical structures between strand 1 and strand 2. We have partially achieved this by extending the dictionary of Bethe/Gauge correspondence: a remarkable relation between quantum integrable systems and supersymmetric gauge theories. This provided a new interpretation of the R-matrices, which are the building blocks of every integrable model, and led to a novel presentation of the Yang-Baxter equation.